Time Value Of Money Homeworkproblem 1
Calculate The Future Value Of A
Calculate the future value of an investment, given the following characteristics: (a) PV: $30,000, (b) NPER: 25, (c) Rate: 5%. Using the information from Problem 1, calculate the future value of the same investment using daily compounding (divide the rate by 365 and multiply the number of years by 365).
Calculate the present value of an investment, given: (a) FV: $120,000, (b) Rate: 14%, and (c) NPER: 20.
Determine the rate of return on an investment, given: (a) FV: $50,000, (b) PV: $1,000, and (c) NPER: 15. Find the number of periods (NPER) for an investment, given: (a) FV: $25,000, (b) PV: $10,000, and (c) Rate: 10%. Calculate the future value (FV) on an ordinary annuity, given: (a) PMT: $1,000, (b) NPER: 15, and (c) Rate: 10%.
Calculate the present value (PV) of an annuity due, given: (a) PMT: $4,000, (b) Rate: 5%, and (c) NPER: 10. For extension population growth: If the current population of city X is 1,000,000 citizens, how long would it take for the city to reach 2,000,000 citizens if the growth rate was 5%?
Paper For Above instruction
The concept of time value of money (TVM) is fundamental to financial decision-making, underpinning many calculations related to investments, loans, and savings. TVM reflects the idea that money available today is worth more than the same amount in the future due to its potential earning capacity. This principle enables investors and financial managers to compare cash flows at different times and to determine the present or future value of investments, loans, or cash flows. This paper discusses key TVM calculations including future value, present value, rate of return, number of periods, annuities, and population growth as applications of the TVM concept.
Calculating Future Value of an Investment
The future value (FV) of an investment can be computed using the formula: FV = PV × (1 + r)^n, where PV is the present value, r is the interest rate per period, and n is the number of periods. For example, considering an initial deposit of $30,000, a 25-year investment horizon, and an annual interest rate of 5%, the FV is calculated as:
FV = $30,000 × (1 + 0.05)^25 ≈ $30,000 × 3.386 ≈ $101,580.
This calculation demonstrates how a modest investment can grow significantly over time. When compounding occurs more frequently, such as daily, the effective interest rate per period becomes r/365,
and n is multiplied by 365. For example, with daily compounding:
FV = PV × (1 + r/365)^(n×365)
Applying the same data:
FV = $30,000 × (1 + 0.05/365)^(25×365) ≈ $30,000 × (1.0001369)^{9125} ≈ $30,000 × 3.439 ≈ $103,170.
Calculating Present Value
Present value (PV) calculations determine how much a future sum is worth today, based on a specified rate. The formula is PV = FV / (1 + r)^n. For example, determining the present value of $120,000 due in 20 years at 14%:
PV = $120,000 / (1 + 0.14)^20 ≈ $120,000 / 10.94 ≈ $10,964.
Determining Rate of Return
The rate of return (r) on an investment can be found by rearranging the FV formula:
r = (FV / PV)^{1/n} - 1.
Given FV = $50,000, PV = $1,000, and NPER = 15:
r = ($50,000 / $1,000)^{1/15} - 1 ≈ 50^{0.0667} - 1 ≈ 1.532 - 1 ≈ 0.532 or 53.2%.
Calculating Number of Periods (NPER)
NPER can be derived from the FV formula by rearrangement:
n = log(FV / PV) / log(1 + r).
For FV = $25,000, PV = $10,000, and r = 10%:
n = log(2.5) / log(1.10) ≈ 0.3979 / 0.0414 ≈ 9.61 periods.
Future Value of an Ordinary Annuity
The FV of an ordinary annuity can be calculated as:
FV = P × [(1 + r)^n - 1] / r,
Where P is the payment per period. Using P = $1,000, r = 10%, and n = 15:
FV = 1,000 × [(1 + 0.10)^15 - 1] / 0.10 ≈ 1,000 × (4.177 - 1) / 0.10 ≈ 1,000 × 31.77 ≈ $31,770.
Present Value of an Annuity Due
An annuity due involves payments made at the beginning of each period. Its PV can be found by multiplying the PV of an ordinary annuity by (1 + r):
PV due = PV ordinary × (1 + r).
For example, with P = $4,000, r = 5%, and n = 10:
PV ordinary = 4,000 × [1 - (1 + 0.05)^{-10}] / 0.05 ≈ 4,000 × 7.7217 ≈ $30,887.
Therefore, PV due = $30,887 × 1.05 ≈ $32,431.
Population Growth Calculation
Population growth can be modeled using the compound growth formula:
P = P0 × (1 + g)^t,
Where P0 is the initial population, g is the growth rate, and t is time in years. To find t when P reaches 2 million:
2,000,000 = 1,000,000 × (1 + 0.05)^t
2 = (1.05)^t
t = log(2) / log(1.05) ≈ 0.3010 / 0.0212 ≈ 14.2 years.
Thus, it will take approximately 14.2 years for the population to double at a 5% annual growth rate.
Conclusion
The applications of the time value of money are crucial in evaluating various financial scenarios, enabling informed decision-making. Whether calculating how investments grow, determining the present worth of future cash flows, or projecting population growth, the fundamental formulas provide valuable insight into financial planning and analysis. Mastering these calculations can empower individuals and organizations to optimize their financial strategies effectively.
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